In how many ways can the letters of the word DUMDUMDIGADIGA be arranged?

Options

A. a. 180
B. b. 90
C. c. 270
D. d. none of these

Answer

To find the number of ways to arrange the letters of the word "DUMDUMDIGADIGA", we will follow these steps:

### Step 1: Count the Total Number of Letters
The word "DUMDUMDIGADIGA" consists of 14 letters.

### Step 2: Identify the Frequency of Each Letter
Next, we need to count how many times each letter appears in the word:
- D appears 4 times
- U appears 2 times
- M appears 2 times
- I appears 2 times
- G appears 2 times
- A appears 2 times

### Step 3: Use the Formula for Arrangements with Repetitions
The formula to find the number of arrangements of letters when there are repetitions is given by:

\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times p_3! \times \ldots} \]

where:
- \( n \) is the total number of letters,
- \( p_1, p_2, p_3, \ldots \) are the frequencies of the repeated letters.

In our case:
- \( n = 14 \)
- The frequencies are: 4 (for D), 2 (for U), 2 (for M), 2 (for I), 2 (for G), and 2 (for A).

Thus, we can write:

\[ \text{Number of arrangements} = \frac{14!}{4! \times 2! \times 2! \times 2! \times 2! \times 2!} \]

### Step 4: Calculate the Factorials
Now, we need to calculate the factorials:
- \( 14! = 87,178,291,200 \)
- \( 4! = 24 \)
- \( 2! = 2 \)

### Step 5: Substitute and Simplify
Now we can substitute these values into our formula:

\[ \text{Number of arrangements} = \frac{87,178,291,200}{24 \times 2 \times 2 \times 2 \times 2 \times 2} \]

Calculating the denominator:

\[ 24 \times 2^5 = 24 \times 32 = 768 \]

Now substituting this back:

\[ \text{Number of arrangements} = \frac{87,178,291,200}{768} \]

### Step 6: Perform the Division
Now we perform the division:

\[ \text{Number of arrangements} = 113,513,400 \]

### Final Answer
Thus, the total number of ways to arrange the letters of the word "DUMDUMDIGADIGA" is **113,513,400**.

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